Properties

Label 4025.2.a.r
Level $4025$
Weight $2$
Character orbit 4025.a
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.122821.1
Defining polynomial: \( x^{5} - 2x^{4} - 4x^{3} + 4x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 805)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{4} + \beta_{3}) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{6} + q^{7} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{8} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{4} + \beta_{3}) q^{4} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{6} + q^{7} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{8} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{9} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{11} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{12} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{13} + ( - \beta_1 + 1) q^{14} + ( - \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 1) q^{16} + ( - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 2) q^{17} + (\beta_{4} + 4 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{18} + ( - \beta_{4} - 3 \beta_1 + 2) q^{19} + (\beta_{3} + 1) q^{21} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{22} - q^{23} + (3 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{24} + (\beta_{4} - \beta_{3} - 4 \beta_1 + 3) q^{26} + (\beta_{4} + \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 4) q^{27} + (\beta_{4} + \beta_{3}) q^{28} + ( - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2) q^{29} + (2 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 - 2) q^{31} + ( - \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{32} + ( - 2 \beta_{4} - 5 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{33} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 1) q^{34} + (3 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 5) q^{36} + ( - 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{37} + (2 \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 4) q^{38} + (3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{39} + (2 \beta_{4} + 4 \beta_1 - 1) q^{41} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{42} + ( - \beta_{4} + \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 1) q^{43} + ( - 3 \beta_{4} - 7 \beta_{3} + 2 \beta_{2} + \beta_1) q^{44} + (\beta_1 - 1) q^{46} + ( - 2 \beta_{4} - \beta_{2} - \beta_1 + 6) q^{47} + (2 \beta_{3} - \beta_{2} - 4 \beta_1 + 2) q^{48} + q^{49} + ( - 4 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - \beta_1 - 3) q^{51} + (\beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 7) q^{52} + (2 \beta_{4} + \beta_{2} + \beta_1 + 3) q^{53} + (5 \beta_{4} + 8 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 7) q^{54} + (\beta_{4} + \beta_{3} - \beta_{2}) q^{56} + ( - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 6) q^{57} + ( - 3 \beta_{4} + \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{58} + ( - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1) q^{59} + (3 \beta_{2} + \beta_1 + 1) q^{61} + (3 \beta_{4} + 2 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{62} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{63} + ( - \beta_{4} + 2 \beta_{2} + \beta_1 - 3) q^{64} + ( - 3 \beta_{4} - 7 \beta_{3} + 6 \beta_{2} + 9 \beta_1 - 10) q^{66} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 6) q^{67} + ( - \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + \beta_1 - 8) q^{68} + ( - \beta_{3} - 1) q^{69} + (2 \beta_{4} + 4 \beta_{3} + \beta_{2} + 5 \beta_1 - 10) q^{71} + (5 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - 5 \beta_1 + 6) q^{72} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{73} + ( - 4 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{74} + (2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5 \beta_1 + 3) q^{76} + ( - \beta_{3} + 2 \beta_{2} - 2) q^{77} + (2 \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 7 \beta_1 + 3) q^{78} + (2 \beta_{4} - \beta_{3} + \beta_{2} + 5 \beta_1 + 1) q^{79} + (5 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 6 \beta_1 + 4) q^{81} + ( - 2 \beta_{4} - 4 \beta_{3} - 5 \beta_1 - 3) q^{82} + ( - \beta_{4} - \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 2) q^{83} + (2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{84} + ( - 5 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{86} + ( - 7 \beta_{4} - 4 \beta_{3} + \beta_{2} - 2 \beta_1 + 8) q^{87} + ( - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 7) q^{88} + ( - 2 \beta_{4} + 5 \beta_{3} - 2 \beta_{2}) q^{89} + (2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{91} + ( - \beta_{4} - \beta_{3}) q^{92} + (3 \beta_{4} + \beta_{3} - 5 \beta_{2} - 4 \beta_1 + 4) q^{93} + ( - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{94} + ( - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 + 5) q^{96} + (5 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{97} + ( - \beta_1 + 1) q^{98} + ( - 5 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 3 q^{2} + 6 q^{3} + 3 q^{4} + 7 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 3 q^{2} + 6 q^{3} + 3 q^{4} + 7 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9} - 11 q^{11} + 14 q^{12} + 7 q^{13} + 3 q^{14} - q^{16} - q^{17} + 17 q^{18} + 2 q^{19} + 6 q^{21} - 2 q^{22} - 5 q^{23} + 12 q^{24} + 8 q^{26} + 15 q^{27} + 3 q^{28} - 14 q^{29} - 6 q^{31} + 4 q^{32} - 22 q^{33} + 4 q^{34} + 28 q^{36} + q^{37} + 31 q^{38} + 15 q^{39} + 7 q^{41} + 7 q^{42} + 12 q^{43} - 11 q^{44} - 3 q^{46} + 24 q^{47} + 4 q^{48} + 5 q^{49} - 28 q^{51} + 36 q^{52} + 21 q^{53} + 49 q^{54} + 3 q^{56} + 15 q^{57} - 9 q^{58} - q^{59} + 7 q^{61} + 26 q^{62} + 5 q^{63} - 15 q^{64} - 45 q^{66} + 35 q^{67} - 43 q^{68} - 6 q^{69} - 32 q^{71} + 36 q^{72} + 7 q^{73} - 10 q^{74} + 10 q^{76} - 11 q^{77} + 8 q^{78} + 18 q^{79} + 21 q^{81} - 33 q^{82} + q^{83} + 14 q^{84} - 26 q^{86} + 18 q^{87} - 41 q^{88} + q^{89} + 7 q^{91} - 3 q^{92} + 19 q^{93} + 14 q^{94} + 26 q^{96} + 9 q^{97} + 3 q^{98} - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 4x^{3} + 4x^{2} + 3x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 2\nu + 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{4} + 2\nu^{3} + 4\nu^{2} - 4\nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{4} + 2\beta_{3} + \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{4} + 8\beta_{3} + 2\beta_{2} + 18\beta _1 + 2 \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.79802
1.17316
0.266708
−0.787589
−1.45030
−1.79802 1.59027 1.23287 0 −2.85933 1.00000 1.37931 −0.471046 0
1.2 −0.173158 −1.11762 −1.97002 0 0.193524 1.00000 0.687440 −1.75093 0
1.3 0.733292 2.28713 −1.46228 0 1.67714 1.00000 −2.53886 2.23098 0
1.4 1.78759 −0.0742225 1.19548 0 −0.132679 1.00000 −1.43816 −2.99449 0
1.5 2.45030 3.31444 4.00395 0 8.12135 1.00000 4.91027 7.98549 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4025.2.a.r 5
5.b even 2 1 805.2.a.k 5
15.d odd 2 1 7245.2.a.bi 5
35.c odd 2 1 5635.2.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.k 5 5.b even 2 1
4025.2.a.r 5 1.a even 1 1 trivial
5635.2.a.x 5 35.c odd 2 1
7245.2.a.bi 5 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\( T_{2}^{5} - 3T_{2}^{4} - 2T_{2}^{3} + 10T_{2}^{2} - 4T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{5} - 6T_{3}^{4} + 8T_{3}^{3} + 7T_{3}^{2} - 13T_{3} - 1 \) Copy content Toggle raw display
\( T_{11}^{5} + 11T_{11}^{4} + 14T_{11}^{3} - 185T_{11}^{2} - 612T_{11} - 452 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 3 T^{4} - 2 T^{3} + 10 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{5} - 6 T^{4} + 8 T^{3} + 7 T^{2} + \cdots - 1 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} + 11 T^{4} + 14 T^{3} + \cdots - 452 \) Copy content Toggle raw display
$13$ \( T^{5} - 7 T^{4} - 15 T^{3} + 146 T^{2} + \cdots - 761 \) Copy content Toggle raw display
$17$ \( T^{5} + T^{4} - 60 T^{3} + 15 T^{2} + \cdots - 1444 \) Copy content Toggle raw display
$19$ \( T^{5} - 2 T^{4} - 45 T^{3} + 153 T^{2} + \cdots - 452 \) Copy content Toggle raw display
$23$ \( (T + 1)^{5} \) Copy content Toggle raw display
$29$ \( T^{5} + 14 T^{4} - 39 T^{3} + \cdots - 7448 \) Copy content Toggle raw display
$31$ \( T^{5} + 6 T^{4} - 70 T^{3} + \cdots + 2489 \) Copy content Toggle raw display
$37$ \( T^{5} - T^{4} - 168 T^{3} + \cdots - 13628 \) Copy content Toggle raw display
$41$ \( T^{5} - 7 T^{4} - 70 T^{3} + 146 T^{2} + \cdots - 331 \) Copy content Toggle raw display
$43$ \( T^{5} - 12 T^{4} - 77 T^{3} + \cdots - 8236 \) Copy content Toggle raw display
$47$ \( T^{5} - 24 T^{4} + 196 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$53$ \( T^{5} - 21 T^{4} + 142 T^{3} + \cdots + 548 \) Copy content Toggle raw display
$59$ \( T^{5} + T^{4} - 70 T^{3} - 131 T^{2} + \cdots + 176 \) Copy content Toggle raw display
$61$ \( T^{5} - 7 T^{4} - 46 T^{3} + \cdots - 2252 \) Copy content Toggle raw display
$67$ \( T^{5} - 35 T^{4} + 450 T^{3} + \cdots + 596 \) Copy content Toggle raw display
$71$ \( T^{5} + 32 T^{4} + 264 T^{3} + \cdots - 54361 \) Copy content Toggle raw display
$73$ \( T^{5} - 7 T^{4} - 89 T^{3} + \cdots + 4297 \) Copy content Toggle raw display
$79$ \( T^{5} - 18 T^{4} - 19 T^{3} + \cdots - 11236 \) Copy content Toggle raw display
$83$ \( T^{5} - T^{4} - 204 T^{3} + \cdots + 10204 \) Copy content Toggle raw display
$89$ \( T^{5} - T^{4} - 230 T^{3} + \cdots - 16204 \) Copy content Toggle raw display
$97$ \( T^{5} - 9 T^{4} - 271 T^{3} + \cdots - 4948 \) Copy content Toggle raw display
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